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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A practical guide to the basic physics that radiation protection professionals need
A much-needed working resource for health physicists and other radiation protection professionals, this volume presents clear, thorough, up-to-date explanations of the basic physics necessary to address real-world problems in radiation protection. Designed for readers with limited as well as basic science backgrounds, Physics for Radiation Protection emphasizes applied concepts and carefully illustrates all topics through examples as well as practice problems.
Physics for Radiation Protection draws substantially on current resource data available for health physics use, providing decay schemes and emission energies for approximately 100 of the most common radionuclides encountered by practitioners. Excerpts of the Chart of the Nuclides, activation cross sections, fission yields, fission-product chains, photon attenuation coefficients, and nuclear masses are also provided. Coverage includes:
- The atom as an energy system
- An overview of the major discoveries in radiation physics
- Extensive discussion of radioactivity, including sources and materials
- Nuclear interactions and processes of radiation dose
- Calculational methods for radiation exposure, dose, and shielding
- Nuclear fission and production of activation and fission products
- Specialty topics ranging from nuclear criticality and applied statistics to X rays
- Extensive and current resource data cross-referenced to standard compendiums
- Extensive appendices and more than 400 figures
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Contents
Preface
1 Structure of Atoms
1.1 Atom Constituents
1.2 Structure, Identity, and Stability of Atoms
1.3 Chart of the Nuclides
1.4 Nuclear Models
Problems – Chapter 1
2 Atoms and Energy
2.1 Atom Measures
2.2 Energy Concepts for Atoms
2.3 Summary
Other Suggested Sources
Problems – Chapter 2
3 Radioactive Transformation
3.1 Processes of Radioactive Transformation
3.2 Decay Schemes
3.3 Rate of Radioactive Transformation
3.4 Radioactivity Calculations
3.5 Activity–mass Relationships
3.6 Radioactive Series Transformation
3.7 Radioactive Equilibrium
3.8 Total Number of Transformations (Uses of τ and λEff)
3.9 Discovery of the Neutrino
Acknowledgments
Other Suggested Sources
Problems – Chapter 3
4 Interactions
4.1 Production of X-rays
4.2 Characteristic X-rays
4.3 Nuclear Interactions
4.4 Alpha Particle Interactions
4.5 Transmutation by Protons and Deuterons
4.6 Neutron Interactions
4.7 Activation Product Calculations
4.8 Medical Isotope Reactions
4.9 Transuranium Elements
4.10 Photon Interactions
4.11 Fission and Fusion Reactions
4.12 Summary
Other Suggested Sources
Problems – Chapter 4
5 Nuclear Fission and its Products
5.1 Fission Energy
5.2 Physics of Sustained Nuclear Fission
5.3 Neutron Economy and Reactivity
5.4 Nuclear Power Reactors
5.5 Light Water Reactors (LWRs)
5.6 Heavy Water Reactors (HWRs)
5.7 Breeder Reactors
5.8 Gas-cooled Reactors
5.9 Reactor Radioactivity
5.10 Radioactivity in Reactors
5.11 Summary
Acknowledgments
Other Suggested Sources
Problems – Chapter 5
6 Naturally Occurring Radiation and Radioactivity
6.1 Discovery and Interpretation
6.2 Background Radiation
6.3 Cosmic Radiation
6.4 Cosmogenic Radionuclides
6.5 Naturally Radioacitve Series
6.6 Singly Occurring Primordial Radionuclides
6.7 Radioactive Ores and Byproducts
6.8 Radioactivity Dating
6.9 Radon and its Progeny
6.10 Summary
Acknowledgements
Other Suggested Sources
Problems – Chapter 6
7 Interactions of Radiation with Matter
7.1 Radiation Dose and Units
7.2 Radiation Dose Calculations
7.3 Interaction Processes
7.4 Interactions of Alpha Particles and Heavy Nuclei
7.5 Beta Particle Interactions and Dose
7.6 Photon Interactions
7.7 Photon Attenuation and Absorption
7.8 Energy Transfer and Absorption by Photons
7.9 Exposure/Dose Calculations
7.10 Summary
Acknowledgments
Other Suggested Sources
Problems – Chapter 7
8 Radiation Shielding
8.1 Shielding of Alpha-Emitting Sources
8.2 Shielding of Beta-Emitting Sources
8.3 Shielding of Photon Sources
8.4 Gamma Flux for Distributed Sources
8.5 Shielding of Protons and Light Ions
8.6 Summary
Acknowledgments
Other Suggested Sources
Problems – Chapter 8
9 Internal Radiation Dose
9.1 Absorbed Dose in Tissue
9.2 Accumulated Dose
9.3 Factors In The Internal Dose Equation
9.4 Radiation Dose from Radionuclide Intakes
9.5 Operational Determinations of Internal Dose
9.6 Tritium: a Special Case
9.7 Summary
Other Suggested Sources
Problems – Chapter 9
10 Environmental Dispersion
10.1 Atmospheric Dispersion
10.2 Nonuniform turbulence: Fumigation, Building Effects
10.3 Puff Releases
10.4 Sector-Averaged χ/Q Values
10.5 Deposition/Depletion: Guassian Plumes
10.6 Summary
Other Suggested Sources
Problems – Chapter 10
11 Nuclear Criticality
11.1 Nuclear Reactors and Criticality
11.2 Nuclear Explosions
11.3 Criticality Accidents
11.4 Radiation Exposures in Criticality Events
11.5 Criticality Safety
11.6 Fission Product Release in Criticality Events
11.7 Summary
Acknowledgments
Other Suggested Sources
Problems – Chapter 11
12 Radiation Detection and Measurement
12.1 Gas-Filled Detectors
12.2 Crystalline Detectors/Spectrometers
12.3 Semiconducting Detectors
12.4 Gamma Spectroscopy
12.5 Portable Field Instruments
12.6 Personnel Dosimeters
12.7 Laboratory Instruments
Other Suggested Sources
Problems – Chapter 12
13 Statistics in Radiation Physics
13.1 Nature of Counting Distributions
13.2 Propagation of Error
13.3 Comparison of Data Sets
13.4 Statistics for the Counting Laboratory
13.5 Levels of Detection
13.6 Minimum Detectable Concentration or Contamination
13.7 Log Normal Data Distributions
Acknowledgment
Other Suggested Sources
Chapter 13 – Problems
14 Neutrons
14.1 Neutron Sources
14.2 Neutron Parameters
14.3 Neutron Interactions
14.4 Neutron Dosimetry
14.5 Neutron Shielding
14.6 Neutron Detection
14.7 Summary
Acknowledgment
Other Suggested Sources
Problems – Chapter 14
Answers to Selected Problems
Appendix A
Appendix B
Appendix C
Appendix D
Index
Related Titles
Turner, J. E., Downing, D. J., Bogard, J. S.
Statistical Methods in Radiation Physics
2012ISBN: 978-3-527-41107-8
Turner, J. E.
Atoms, Radiation, and Radiation Protection
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Knoll, G. F.
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Bevelacqua, J. J.
Basic Health Physics
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1999ISBN: 978-0-471-29711-6
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Contemporary Health Physics
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1995ISBN: 978-0-471-01801-8
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Introduction to Radiological Physics and Radiation Dosimetry
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The Author
James E. Martin2604 Bedford RoadAnn Arbour MI 48104USA
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© 2013 Wiley-VCH Verlag & Co. KGaA, Boschstr. 12, 69469 Weinheim, germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
ePDF ISBN: 978-3-527-66709-3ePub ISBN: 978-3-527-66708-6mobi ISBN: 978-3-527-66707-9oBook ISBN: 978-3-527-66706-2
To the memory of
Frank A. and Virginia E. Martin and JoAnn Martin Burkhart.
Preface
This book is the outcome of teaching radiation physics to students beginning a course of study in radiation protection, or health physics. This 3rd edition attempts as the first two did to provide in one place a comprehensive treatise of the major physics concepts required of radiation protection professionals. Numerous real-world examples and practice problems are provided to demonstrate concepts and hone skills, and even though its limited uses are thoroughly developed and explained, some familiarity with calculus would be helpful in grasping some of the subjects.
The materials in this compendium can be used in a variety of ways, both for instruction and reference. The first two chapters describe the atom as an energy system, and as such they may be of most use for those with minimal science background. Chapter 3 addresses the special condition of radioactive transformation (or disintegration) of atoms with excess energy, regardless of how acquired. Chapters 4 and 5 describe activation and fission processes and the amount of energy gained or lost due to atom changes; these define many of the sources that are addressed in radiation protection. Chapter 6 develops natural sources of radiation and radioactive materials primarily as reference material; however, the sections on radioactive dating and radon could be used as supplemental, though specialized, material to Chapter 3.
The interaction of radiation with matter and the resulting deposition of energy is covered in Chapter 7 along with the corollary subjects of radiation exposure and dose. Radiation shielding, also related to interaction processes, is described in Chapter 8 for various source geometries. Chapters 9 and 10, on internal radiation dose and environmental dispersion of radioactive materials, are also fundamental for understanding how such materials produce radiation dose inside the body and how they become available for intakes by humans. These are followed by specialty chapters on nuclear criticality (Chapter 11); radiation detection and measurement (Chapter 12); applied statistics (Chapter 13); and finally (Chapter 14) neutron sources and interactions. A course in radiation physics would likely include the material in Chapters 3, 4, 5, 7, and 8 with selections from the other chapters, all or in part, to develop needed background and to address specialty areas of interest to instructor and student. In anticipation of such uses, attempts have been made to provide comprehensive and current coverage of the material in each chapter and relevant data sets.
Health physics problems require resource data. To this end, decay schemes and associated radiation emissions are included for about 100 of the most common radionuclides encountered in radiation protection. These are developed in the detail needed for health physics uses and cross referenced to standard compendiums for straightforward use when these more in-depth listings need to be consulted. Resources are also provided on activation cross sections, fission yields, fission-product chains, photon absorption coefficients, nuclear masses, and abbreviated excerpts of the Chart of the Nuclides. These are current from the National Nuclear Data Center at Brookhaven National Laboratory; the Center and its staff are a national resource.
The units used in radiation protection have evolved over the hundred years or so that encompass the field. They continue to do so with a fairly recent, but not entirely accepted, emphasis on System Internationale (SI) Units while U.S. standards and regulations have continued to use conventional units. To the degree possible, this book uses fundamental quantities such as eV, transformations, time, distance, and the numbers of atoms or emitted particles and radiations to describe nuclear processes, primarily because they are basic to concepts being described but partially to avoid conflict between SI units and conventional ones. Both sets of units are defined as they apply to radiation protection, but in general the more fundamental parameters are used. For the specific units of radiation protection such as exposure, absorbed dose, dose equivalent, and activity, text material and examples are generally presented in conventional units because the field is very much an applied one; however, the respective SI unit is also included where feasible. By doing so, it is believed presentations are clearer and relevant to the current conditions, but it is recognized that this quandary is likely to continue.
This endeavor has been possible because of the many contributions of my research associates and students whose feedback shaped the teacher on the extent and depth of the physics materials necessary to function as a professional health physicist. I am particularly indebted to Chul Lee who began this process with me with skill and patience and to Rachael Nelson who provided invaluable help in capping off this 3rd edition. I hope it helps all who undertake study in this exciting field to appreciate how physics underpins it.
In an undertaking of this scope, it is inevitable that undetected mistakes creep in and remain despite the best efforts of preparers and editors; thus, reports ([email protected]) of errors found would be appreciated.
James E. Martin, Ph.D., CHPAssociate Professor (emeritus) ofRadiological HealthThe University of Michigan, 2012
1Structure of Atoms
“I have discovered something
very interesting.”
W. C. Roentgen (Nov. 8, 1895)
The fifty years following Roentgen’s discovery of x-rays saw remarkable changes in physics that literally changed the world forever, culminating in a host of new products from nuclear fission. Discovery of the electron (1897) and radioactivity (1898) focused attention on the makeup of atoms and their structure as did other discoveries. For example, in 1900, Planck introduced the concept that the emission (or absorption) of electromagnetic radiation occurs only in a discrete amount where the energy is proportional to the frequency, v, of the radiation, or
where h is a constant (Planck’s constant) of nature; its value is
Planck presumed that he had merely found an ad hoc solution for blackbody radiation, but in fact he had discovered a basic law of nature: any physical system capable of emitting or absorbing electromagnetic radiation is limited to a discrete set of possible energy values or levels; energies intermediate between them simply do not occur. Planck’s theory was revolutionary because it states that the emission and absorption of radiation must be discontinuous processes, i.e. only as a transition from one particular energy state to another where the energy difference is an integral multiple of hv. This revolutionary theory extends over 22 orders of magnitude from very long wavelength radiation such as radio waves up to and including high energy gamma rays. It includes the energy states of particles in atoms and greatly influences their structure.
Einstein (in 1905) used Planck’s discrete emissions (or quanta) to explain why light of a certain frequency (wavelength) causes the emission of electrons from the surface of various metals (the photoelectric effect). Light photons clearly have no rest mass, but behaving like a particle, a photon can hit a bound electron and “knock” it out of the atom.
Example 1–1. Light with a wavelength of 5893 Å produces electrons from a potassium surface that are stopped by 0.36 volts. Determine: a) the maximum energy of the photoelectron, and b) the work function.
Solution. a) The maximum energy KEmax of the ejected photoelectrons is equal to the stopping potential of 0.36 eV.
b) the work function is the energy of the incident photon minus the energy given to the ejected electron, or
A. H. Compton used a similar approach to explain x-ray scattering as interactions between “particle-like” photons and loosely bound (or “free”) electrons of carbon (now known as the Compton effect). Energy and momentum are conserved and the calculated wavelength changes agreed with experimental observations.
1.1Atom Constituents
Atoms consist of protons and neutrons (discovered in 1932 by Chadwick) bound together to form a nucleus which is surrounded by electrons that counterbalance each proton in the nucleus to form an electrically neutral atom. Its components are: a) protons which have a reference mass of about 1.0 and an electrical charge of +1; b) electrons which have a mass about 1/1840 of the proton and a (–1) electrical charge; and c) neutrons which are electrically neutral and slightly heavier than the proton. The number of protons (or Z) establishes the identity of the atom and its mass number (A) is the sum of protons and neutrons (or N) in its nucleus. Electrons do not, and, according to the uncertainty principle, cannot exist in the nucleus, although they can be manufactured and ejected during radioactive transformation. Modern theory has shown that protons and neutrons are made up of quarks, leptons, and bosons (recently discovered), but these are not necessary for understanding atoms or how they produce radiant energy.
Four forces of nature determine the array of atom constituents. The electromagnetic force between charged particles is attractive if the charges (q1 and q2) are of opposite signs (i.e. positive or negative); if of the same sign, the force F will be repulsive and quite strong for the small distances between protons in the nucleus of an atom. This repulsion is overcome by the nuclear force (or strong force) which is about 100 times stronger; it only exists in the nucleus and only between protons and neutrons (there is no center point towards which nucleons are attracted). The weak force (relatively speaking) has been shown to be a form of the electromagnetic force; it influences radioactive transformation; and the gravitational force, though present, is negligible in atoms.
The nucleus of an atom containing Z protons is essentially a charged particle (with charge Ze) that attracts an equal number of electrons that orbit the nucleus some distance away. Thomson theorized that each negatively charged electron was offset by a positively charged proton and that these were arrayed somewhat like a plum pudding to form an electrically neutral atom. This model proved unsatisfactory for explaining the large-angle scattering of alpha particles by gold foils as observed by Rutherford and Geiger-Marsden. Such large deflections were due the electromagnetic force between a positively charged nucleus (Ze) at the center of the atom and that of the alpha particle (2e).
The force for such deflections is inversely proportional to the distance r between them, or
which yields a value of r of about 10–15 m which Rutherford proposed as the radius of a small positively-charged nucleus surrounded by electrons in orbits about 10–10 m in size. This model had a fatal flaw: according to classical physics the electrons would experience acceleration, v2/r, causing them to continuously emit radiation and to quickly (in about 10–8 s) spiral into the nucleus.
In 1913, Niels Bohr explained Rutherford’s conundrum by simply declaring (postulate I) that atoms are stable and that an electron in its orbit does not radiate energy, but only does so when it experiences one of Planck’s quantum changes to an orbit of lower potential energy (postulate III) with the emission of a photon of energy
And, that the allowed stationary states for orbiting electrons (postulate II) are those for which the orbital angular momentum, L, is an integral multiple of h/2π, or:
Bohr assumed that electrons orbiting a nucleus moved in circular orbits under the influence of two force fields: the coulomb attraction (a centripetal force) provided by the positively charged nucleus and the centrifugal force of each electron in orbital motion at a radius, rn, and velocity, mn. These forces are equal and opposite each other, or:
where rn can be calculated from postulate II. And, since q1 and q2 are unity for hydrogen
These relationships can be used to calculate the total energy En of an electron in the nth orbit where the sum of its kinetic and potential energy is
which is the binding energy of the electron in hydrogen and is in perfect agreement with the measured value of the energy required to ionize hydrogen. For other values of n, the allowed energy levels of hydrogen are:
In 1926, Louis de Broglie postulated that if Einstein’s and Compton’s assignment of particle properties to waves was correct, why shouldn’t the converse be true; i.e., that particles have wave properties such that an electron (or a car for that matter) has a wavelength, associated with its motion, or
and that as a wave it has momentum, p, with the value:
This simple but far-reaching concept was later proved by Davisson and Germer who observed diffraction (a wave phenomenon) of electrons (clearly particles) from a nickel crystal. De Broglie’s wave/particle behavior of electrons also opened the door to description of the dynamics of particles by wave mechanics, perhaps the most revolutionary development in physics since Einstein’s special theory of relativity.
1.2Structure, Identity, and Stability of Atoms
The identity of an atom is determined by the number and array of protons and neutrons in its nucleus. An atom with one proton is defined as hydrogen; it has one orbital electron for electrical neutrality. Deuterium (or hydrogen-2) also contains one proton and one electron but also a neutron and is quite stable; tritium (hydrogen-3) with one proton and one electron has a second neutron which causes it to be unstable, or radioactive. These three are isotopes of hydrogen.
Two protons cannot be joined to form an atom because the repulsive electromagnetic force between them is so great that it even overcomes the strongly attractive nuclear force. If, however, a neutron is present, the distribution of forces is such that a stable nucleus is formed and two electrons will then join up to balance the two plus (+) charges of the protons to create a stable, electrically neutral atom of helium so defined because it has two protons. Its mass number (A) is 3 (2 protons plus 1 neutron) and is written as helium-3 or 3He. Because neutrons provide a cozy effect, yet another neutron can be added to obtain 4He which still has two electrons to balance the two positive charges. This atom is the predominant form (or isotope) of helium on earth, and it is very stable (this same atom, minus the two orbital electrons, is ejected from some radioactive atoms as an alpha particle, i.e., a charged helium nucleus). Helium-5 (5He) cannot be formed because the extra neutron creates a very unstable atom that breaks apart very fast (in 10–21 s or so). But, for many atoms an extra neutron(s) is easily accommodated to yield one or more isotopes of the same element, and for some elements adding an extra neutron (or proton) to a nucleus only destabilizes it; i.e., it will often exist as an unstable, or radioactive, atom. Such is the case for hydrogen-3 (3H, or tritium) and carbon-14 (14C). Elements are often identified by name and mass number, e.g., hydrogen-3 (3H) or carbon-14 (14C).
Three protons can be assembled with three neutrons to form lithium-6 (6Li) or with four neutrons, lithium-7 (7Li). Since lithium contains three protons, it must also have three orbital electrons, but because the first orbit can only hold two electrons (there is an important reason for this which is explained by quantum theory) the third electron occupies another orbit further away.
1.3Chart of the Nuclides
As shown in Figure 1-1, a plot of the number of protons versus the number of neutrons increases steadily for heavier atoms because extra neutrons are necessary to distribute the nuclear force and moderate the repulsive electrostatic force between protons. The heaviest element in nature is 238U with 92 protons and 146 neutrons; it is radioactive, but very long-lived. The heaviest stable element in nature is 209Bi with 83 protons and 126 neutrons. Lead with 82 protons is much more common in nature than bismuth and for a long time was thought to be the heaviest of the stable elements; it is also the stable endpoint of the radioactive transformation of uranium and thorium, two primordial naturally occurring radioactive elements (see Chapter 6).
The chart of the nuclides contains basic information on each element, how many isotopes it has (atoms on the horizontal lines) and which ones are stable (shaded) or unstable (unshaded). A good example of such information is shown in Figure 1-2 for four isotopes of carbon (actually there are 8 measured isotopes of carbon but these 4 are the most important). They are all carbon because each contains 6 protons, but each has a different number of neutrons, hence they are distinct isotopes with different weights. 12C and 13C are shaded and thus are stable, as are the two shaded blocks for boron (5 protons) and nitrogen (7 protons) also shown in Figure 1-2. The nuclides in the unshaded blocks (e.g., 11C and 14C) are unstable simply because they don’t have the right array of protons and neutrons to be stable (we will use these properties later to discuss radioactive transformation). The dark band at the top of the block for 14C denotes that it is a naturally-occurring radioactive isotope, a convention used for several other such radionuclides. The block to the far left contains information on naturally abundant carbon: it contains the chemical symbol, C, the name of the element, and the atomic weight of natural carbon, or 12.0107 grams/mole, weighted according to the percent abundance of the two naturally occurring stable isotopes. The shaded blocks contain the atom percent abundance of 12C and 13C in natural carbon at 98.90 and 1.10 atom percent, respectively; these are listed just below the chemical symbol. Similar information is provided for all of the elements in the chart of the nuclides.
Fig. 1-1 Part of the chart of the nuclides. (Nuclides and Isotopes, 16th Edition, KAPL, Inc, 2002.)
1.4Nuclear Models
The array of protons and neutrons in each element is unique because nature forces these constituents toward the lowest potential energy possible; when they attain it they are stable, and until they do they have excess energy and are thus unstable, or radioactive; e.g. tritium or carbon-14. Descriptions of the dynamics and changes in energy states of nuclear constituents often use a shell model; however, descriptions of fission and other phenomena are best done with a liquid drop model. The exact form of the nuclear force in the nucleus is not yet known nor the structure of potential energy states of its constituents, but a shell model corresponds nicely with the emission of gamma rays from excited nuclei. These emissions are similar to those that occur when orbital electrons change to one of lower potential energy.
where A is the atomic mass number of the atom in question and the constant ro has an average value of about 1.3 × 10–15 m, or 1.3 fermi.
Fig. 1–3 Very simplified model of a nebular atom consisting of an array of protons and neutrons with shell-like states within a nucleus surrounded by a cloud of electrons with three dimensional wave patterns and also with shell-like energy states.
Problems – Chapter 1
1–1. How many neutrons and how many protons are there in: a) 14C, b) 27Al, c) 133Xe, and d) 209Bi?
1–2. Calculate the radius of the nucleus of 27Al in meters and fermis.
1–3. When light of wavelength 3132 Å falls on a caesium surface, a photoelectron is emitted for which the stopping potential is 1.98 volts. Calculate the maximum energy of the photoelectron, the work function, and the threshold frequency.
1–4. The work function of potassium is 2.20 eV. What should be the wavelength of the incident electromagnetic radiation so that the photoelectrons emitted from potassium will have a maximum kinetic energy of 4 eV? Also calculate the threshold frequency.
1–5. Calculate the de Broglie wavelength associated with the following:
a) an electron with a kinetic energy of 1 eV
b) an electron with a kinetic energy of 510 keV
c) a thermal neutron (2200 m/s)
d) a 1500 kg automobile at a speed of 100 km/h.
1–6. Calculate the de Broglie wavelength associated with: a) a proton with 15 MeV of kinetic energy, and b) a neutron of the same energy.
2Atoms and Energy
A. Einstein (1905)
Atoms contain enormous amounts of energy distributed among the energy states of its constituent parts. A decrease in the potential energy state of one or more of the constituents will cause the emission of energy either as a particle or a photon, or both. For example, an electron “free” of the nucleus will experience a decrease in potential energy as it goes from the surface (the “free” state) to level 2, or on down to level 1.
The levels represent negative energy states because it would take work to return the electrons to an unbound (or “free”) state. The atom with bound electrons has been determined to be slightly lighter, and the loss of mass exactly matches the energy of the emitted photon(s). These concepts also apply to protons and neutrons which are bound in the nucleus at different energy levels, or
2.1Atom Measures
Many radiation protection problems require knowledge of the number of atoms (Avogadro’s number) in an amount (mass) of an element; the mass of each atom and its component particles; and the energy (in electron volts) associated with mass changes in and between atoms. Appendix A contains these and other key parameters related to atoms and radiation physics.
Avogadro’s number, NA, is the number of atoms or molecules in a mole of any substance and is a constant, independent of the nature of the substance. When Avogadro stated the concept in 1811 he had no knowledge of its magnitude only that the number was very large. Its modern value is:
Example 2–1. Calculate the number of atoms of 13C in 0.1 gram of natural carbon.
Solution. From Figure 1–2, the atomic weight of carbon is 12.0107 g and the atom percent abundance of 13C is 1.10%. Thus
The atomic mass unit, or to be precise the unified mass unit, u, is used to express the masses of atom constituents as well as atoms themselves (since the sum of protons and neutrons in individual atoms is the mass number, one of these must be a “mass unit”). The u is defined as one-twelfth the mass of the neutral 12C atom, which weighs exactly 12.000000 g. All other elements and their isotopes are assigned weights relative to 12C. The atomic mass unit was originally defined relative to oxygen-16 at 16.000000 grams per mole but carbon-12 has proved to be a better reference nuclide, and since 1962, atomic masses have been based on the unified mass scale referenced to carbon-12.
The mass of a single atom of 12C can be obtained from the mass of one mole of 12C which contains Avogadro’s number of atoms as follows:
This mass is shared by 6 protons and 6 neutrons; thus, the average mass of each of the 12 building blocks of the carbon-12 atom, including the paired electrons, can be calculated by distributing the mass of one atom of carbon-12 over the 12 nucleons. This quantity, or u, has the value:
which is close to the actual mass of the proton (actually 1.6726231 × 10–24 g) or the neutron (actually 1.6749286 × 10–24 g). In unified mass units, the mass of the proton is 1.00727647 u and that of the neutron is 1.008664923 u. Each of these values is so close to unity that the mass number of an isotope is thus a close approximation of its atomic weight. Measured masses of elements are given in unified mass units, u, and these are listed at the bottom of their respective blocks (e.g., 12C and 13C in Figure 1–2) in the chart of the nuclides. Masses of atoms have been determined to six or more decimal places, and since energy changes in nuclear processes represent mass changes, accurate masses (as listed in Appendix B) are very useful for calculations of energies of nuclear events, e.g. radioactive transformation.
The electron volt (eV) is defined as the increase in kinetic energy of a particle with one unit of electric charge (e.g., an electron) when it is accelerated through a potential difference of one volt, or
Example 2–2. An x-ray tube accelerates electrons from a cathode into a tungsten target anode to produce x-rays. If the electric potential across the tube is 90 kilovolts, what will be the energy of the electrons when they hit the target in eV, joules, ergs?
and in ergs
2.2Energy Concepts for Atoms
The energy associated with atoms is governed by Einstein’s special theory of relativity which states that mass and energy are one and the same. He also showed that the mass m of a body varies with its speed, v, according to
where m0 is the rest mass and c is the velocity of light in a vacuum, which is a constant. The net force on a body is, according to Newton’s second law, a function of its momentum, or
which, if mass can be assumed to remain constant, reduces to
which is the classical relationship used to calculate the force on objects at low speeds (less than 10% of the speed of light). For atoms, however, many particles in and associated with atoms move at high speeds; thus, Newton’s second law must be stated in terms of the relativistic mass, or
In relativistic mechanics, as is classical mechanics, the kinetic energy, KE, of a body is equal to the work done by a force over a distance ds:
If the term in the parenthesis is differentiated, and the integration performed,
thus, the kinetic energy (KE) gained by a moving particle occurs from the mass increase due to its motion; it is also the difference between the total energy, mc2, of the particle and its rest energy, moc2. Thus, mass and energy are equivalent, or
Although Einstein’s concepts are fundamental to atomic phenomena, they are even more remarkable because when he stated them in 1905 no model of the atom existed. He had deduced the theory in search of the basic laws of nature that govern the dynamics and motion of objects. Einstein’s discoveries encompass Newton’s laws for the dynamics of macro-world objects but more importantly also apply to micro-world objects where velocities approach the speed of light; Newton’s laws break down at these speeds, but Einstein’s relationships do not. Einstein believed that the forces of nature were interconnected and he sought, without success, a unified field theory for it.
2.2.1Mass-energy
Energy changes in nuclear processes are readily determined by the mass changes that occur. The energy equivalent of one unified mass unit, u, is
And, for an electron mass:
or 8.988 × 1013 J, which is about 25 million kilowatt . hours, an enormous amount of energy. Both fission and fusion cause atoms to become more tightly bound (see below) with significant changes in mass that is converted to energy.
2.2.2Binding Energy of Nuclei
The potential energy states of the particles that make up an atom are less than when they exist separately; i.e., if two masses, m1 and m2, are brought together to form an atom of mass M, it will hold together only if:
M < m1 + m2
The energy released in binding the two masses is called the binding energy, Eb, or
which is released as photon energy as the constituents become bound to form deuterium. This calculation could also be written as a nuclear reaction
The Q-value represents the amount of energy that is gained when atoms change their bound states or that must be supplied to break them apart in a particular way. For example, in order to break 2H into a proton and a neutron, an energy equal to 2.2246 MeV (the Q-value) would need to be added to the 2H atom, usually as a photon, i.e.
If the photon energy is greater than 2.2246 MeV the excess energy will exist as kinetic energy shared by the proton and the neutron. For tritium, the constituent masses are:
mass of 1 proton
mass of 2 neutrons
mass of electron
Total
which is larger than the measured mass of 3.016049 u (see Appendix B) for 3H by 0.0091061 u; thus, and the total binding energy of tritium is
and the average binding energy for each of its three nucleons is 2.83 MeV per nucleon.
Fig. 2-1 Curve of binding energy per nucleon versus atomic mass number (plotted from data in Appendix B).
2.3Summary
Einstein’s theory of special relativity is applicable to atom systems where particles undergo interchangeable mass/energy processes. These processes yield “radiation” which can be characterized as the emission of energy in the form of particles or electromagnetic energy proportional to the change in mass that occurs as atom constituents change potential energy states. The mass values are very exact as is the calculated energy change (or Q -value). Binding energy is one important result of such calculations; it denotes the amount of energy given off as constituents come together to form an atom, or alternatively, the energy required to disengage them.
Other Suggested Sources
Chart of the nuclides, in Nuclides and Isotopes, 16th Edition, 2002. Available at: www.ChartOfTheNuclides.com.
National Nuclear Data Center, Brookhaven National Laboratory, Upton, Long Island, NY 11973. Data resources are accessible through the internet at www.nndc.bnl.gov.
Problems – Chapter 2
2–2. A one gram target of natural lithium is to be put into an accelerator for bombardment of 6Li to produce 3H. Use the information in Figure 1-1 to calculate the number of atoms of 6Li in the target.
2–3. If one were to base the atomic mass scale on 16O at 16.000000 atomic mass units (amu), calculate the mass of the 16O atom and the mass of one amu. Why is its mass on the 12C scale different from 16.000000?
2–4. Calculate the number of atoms in one gram of natural hydrogen.
2–5. Hydrogen is a diatomic molecule, or H2. Calculate the mass of one molecule of H2.
2–6. Electrons in an x-ray tube are accelerated through a potential difference of 3000 volts. What is the minimum wavelength of the x-rays produced in a target?
2–7. A linear accelerator is operated at 700 kilovolts to accelerate protons. What energy will the protons have when they exit the accelerator in: a) eV, and b) joules. If deuterium ions are accelerated what will be the corresponding energy?
2–8. Calculate the velocity required to double the mass of a particle.
2–9. Use the masses in Appendix B to calculate the total binding energy and the binding energy per nucleon of: a) beryllium-7, b) iron-56, c) nickel-62, and d) uranium-238.
3Radioactive Transformation
“Don’t call it transmutation…they’ll have our heads off as alchemists.”
Ernest Rutherford (1902)
Much of what has been learned about atomic and nuclear physics is based on this remarkable property by which certain nuclei transform themselves spontaneously from one value of Z and N to another. Discovery of the emissions of alpha and beta particles established that atoms are not indivisible, but are made up of more fundamental particles. Use of the emitted particles as projectiles to transform nuclei led eventually to some of the greatest discoveries in physics.
The process of radioactive transformation was recognized by Rutherford as transmutation of one element to another. It is also quite common to use the term radioactive decay, but transformation is a more accurate description of what actually happens; decay suggests a process of disappearance when what actually happens is an atom with excess energy transforms itself to another atom that is either stable or one with more favorable conditions to proceed on to stability.
3.1Processes of Radioactive Transformation
Atoms undergo radioactive transformation because constituents in the nucleus are not arrayed in the lowest potential energy states possible; therefore, a rearrangement of the nucleus occurs in such a way that this excess energy is emitted and the nucleus is transformed to an atom of a new element. The transformation of a nucleus may involve the emission of alpha particles, negatrons, positrons, electromagnetic radiation in the form of x-rays or gamma rays, and, to a lesser extent, neutrons, protons, and fission fragments. Such transformations are spontaneous, and the Q-values are positive; if the array of nuclear constituents is in the lowest potential energy states possible, the transformation yields a stable atom; if not, another transformation must occur.
Fig. 3-1 Line of stable nuclides on the chart of the nuclides relative to neutron-rich (below the line), proton-rich (above the line), and long-lived heavy (Z > 83) nuclei.
Regardless of their origin, radioactive nuclides surround the “zigzag” line of stable nuclei. These can be grouped into three major categories that will determine how they must undergo transformation to become stable:
The basic energy changes represented by these groupings comprise several transformation modes of importance to health physics for they govern how a particular nuclide undergoes transformation and the form of the emitted energy. The characteristics of these emissions in turn establish how much energy is available for potential absorption in a medium to produce biological change(s) and how they may be detected. We examine first the energetics of these transformations, followed by the mathematical laws describing the rate of radioactive transformation.
3.1.1Transformation of Neutron-rich Radioactive Nuclei
Neutron-rich nuclei fall below the line of stable nuclei as shown in Figure 3-2 for 14C, an activation product produced by an (n,p) reaction with stable nitrogen thus reducing the proton number from 7 to 6 and increasing the neutron number from 7 to 8. For neutron-rich nuclei to become stable, they need to reduce the number of neutrons in order to become one of the stable nuclides which are diagonally up and to the left on the chart of the nuclides. In simplest terms, this requires a reduction of one negative charge (or addition of a positive one) in the nucleus. Since these nuclei have excess energy (mass), the transformation can occur if the nucleus emits a negatively charged electron (or beta particle). This results in an increase in the charge on the nucleus and a slight reduction of mass (equal to the electron mass and the mass equivalence of emitted energy) and has the effect of conversion of a neutron to a proton. The negatively charged electron is emitted with high energy. The mass change is small; the only requirement is to reduce the ratio of neutrons to protons.
The transformation of 14C (see Figure 3-2) by beta particle emission can be shown by a reaction equation as follows:
The Q-value for this transformation is 0.156 MeV and is positive, which it must be for the transformation to be spontaneous. This energy is distributed between the recoiling product nucleus (negligible) and the ejected electron, which has most of it. When 14C undergoes transformation to 14N, the atomic number Z increases by 1, the neutron number N decreases by 1, and the mass number A remains the same. The transition should be thought of as a total nuclear change producing a decrease in the neutron number (or an increase in the proton number); this change is shown in Figure 3-2 as a shift upward and to the left on the chart of the nuclides to stable 14N.
Radioactive transformations are typically displayed in decay schemes, which are a diagram of energy (vertical axis) versus Z (horizontal axis). Because the net result is an increase in atomic number and a decrease in total energy, transformation by beta particle (β–) emission is shown by an arrow down and to the right on a plot of energy versus Z, also shown in Figure 3-2.
Fig. 3-2 Position of 14C relative to its transformation product, 14N, on the chart of the nuclides and its decay scheme.
Somewhat more complex transformations of neutron-rich radionuclides are shown in Figure 3-3 for 60Co and 137Cs along with their positions on the chart of the nuclides. These neutron-rich nuclei fall below the line of stability; some are fission products (e.g., 137Cs), but they may also be the product of nuclear interactions (e.g., 60Co, an activation product) which result in an increase in the N number. Both of these radionuclides undergo transformation by the emission of two beta particles with different probabilities, and gamma emission follows one of the routes to relieve the excitation energy that remains because the shell configuration of the neutrons in the nucleus has not achieved the lowest potential energy states possible via the beta emission.
Decay schemes for simplicity do not typically show the vertical and horizontal axes, but it is understood that the transformation product is depicted below the radioactive nuclide (i.e., with less energy) and that the direction of the arrow indicates the change in atomic number Z. For similar reasons, gamma emission (Figure 3-3) is indicated by a vertical line (no change in Z) often with a wave motion.
Fig. 3-3 Artificial radionuclides of neutron-rich 60Co and 137Cs relative to stable nuclides and their transition by β– emission (note that the excerpt from the chart of the nuclides lists a gamma ray energy for 137Cs of 661.7D indicating that it is delayed, which occurs through 137mBa).
Unlike alpha particles, which are monoenergetic from a given source, beta particles are emitted with a range of energies ranging from just above 0 MeV to the maximum energy (denoted as Eβmax) available from the mass change for the particular radioactive nuclide. These energies form a continuous spectrum, as illustrated in Figure 3-4 for 14C.
Fig. 3-4 Continuous spectrum of beta particle emission from 14C.
When it was first observed that beta particles are emitted with a spectrum of energies, it was most perplexing to physicists, since conservation of energy requires that all beta transformations from a given source undergo the same energy change since the mass change is constant. For example, in the transformation of 14C to 14N, the energy of the nucleus decreases by 0.156 MeV because of the decrease in mass that occurs during the transformation. Some of the emitted beta particles have the maximum energy, Eβ,max, but most do not, being emitted at some smaller value with no observable reason since the “missing” energy is not detectable. The “missing” energy posed an enigma to physicists: is the well-established law of conservation of energy wrong, is Einstein’s mass–energy relation wrong, or is there a way to explain the missing energy? The suggestion was made by Pauli and further developed by Fermi in 1934 that the emission of a beta particle is accompanied by the simultaneous emission of another particle, a neutrino (Fermi’s “little one”) with no charge and essentially no mass, that would carry off the rest of the energy. The neutrino (actually an antineutrino, , from 14C transformation to conserve spin) does possess momentum and energy; however, it is so evasive that its existence was not confirmed until 1956 when Reines and Cowan obtained direct evidence for it using an extremely high beta field from a nuclear reactor and an ingenious experiment as described later in this chapter.
Radioactive transformation is a process that involves the whole atom which is demonstrated by the transformation of 187Re to 187Os by negative beta particle emission; i.e., the entire atom must “participate.” The Q-value for 187Re→187Os + β– is 2.64 keV; however, the binding energy of atomic electrons in Os is 15.3 keV greater than in Re. This means that a Re nucleus weighs less than an Os nucleus plus an electron by 12.7 keV (15.3 – 2.6 keV) and the 187Re nucleus alone cannot supply the mass/energy to yield 187Os. The entire 187Re atom must supply the energy for the transformation from 187Re to 187Os by an adjustment of all of the constituent particles and energy states; this circumstance influences the very long half-life of 4.3 × 1010 y of this primordial radionuclide.
3.1.2Double Beta (bb) Transformation
A very rare transformation of neutron-rich radionuclides is ββ transformation, in which two negatively charged beta particles are emitted in cascade. Double beta transformation can occur in cases where only one beta transformation would be energetically impossible. For example, 128Te to 128I would require a Q-value of – 1.26 MeV which cannot occur spontaneously, but ββ transformation from 128Te to 128Xe has a Q-value of 0.87 MeV, and is therefore energetically possible.
Since two beta particles must be emitted for ββ transformation to occur, it would be expected to be highly improbable and the half-life would be very long. A classic example of ββ transformation is 130Te to 130Xe associated with the primordial radionuclide 130Te which has a half-life of 2.5 × 1021 y (see the chart of the nuclides). The long half-life for ββ transformation of 130Te was determined by mass spectroscopy. The method was based on observing an excess abundance of Xe (relative to its abundance in atmospheric Xe) in a tellurium-bearing rock. If the rock is T years old and short compared to the ββ half-life of 130Te, the number of Xe atoms from ββ transformation is
thus the half-life of 130Te is
The number of Te and Xe atoms measured by the mass spectrometer can be used directly to determine the ββ half-life for 130Te → 130Xe, which is 2.5 × 1021 y. Similarly, the ββ half-life for 82Se → 82Kr has been measured at 1.4 × 1020 y.
The long half-lives of ββ transformation preclude direct detection since a mole of a sample would produce just a few transformations per year, and this rate would be virtually impossible to distinguish from natural radioactivity or cosmic rays.
3.1.3Transformation of Proton-rich Nuclei
Proton-rich nuclei have charge and mass such that they are above the line of stable nuclei, i.e., they are unstable due to an excess of protons. These nuclei are typically produced by interactions of protons or deuterons with stable target materials, thereby increasing the number of protons in the target nuclei. Such interactions commonly occur with cyclotrons, linear accelerators, or other particle accelerators, and there are numerous (p,n) reactions that occur in nuclear reactors that produce them. Proton-rich nuclei achieve stability by a total nuclear change in which a positively charged electron mass, or positron, is emitted or an orbital electron is captured by the unstable nucleus. Both processes transform the atom to one with a lower Z value.
Some of the common proton-rich radionuclides, 11C, 13N, 18F, and 22Na, are shown in Figure 3-5 as they appear on the chart of the nuclides relative to stable nuclei. These nuclei will transform themselves to the stable elements 11B, 13C, 18O, and 22Ne, respectively, which are immediately down and to the right (i.e., on the diagonal) on the chart. The mass number does not change in these transitions, only the ratio Z/N.
Fig. 3-5 Proton-rich radionuclides 11C, 13N, 18F, and 22Na relative to stable nuclei and neutron-rich 3H and 14C and the radioactive free neutron.
